LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

Full State Feedback Optimal ControlEdit

Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given.   methods formulate this task as an optimization problem and attempt to minimize the   norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the   we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The SystemEdit

The system is represented using the 9-matrix notation shown below.

 

where   is the state,   is the regulated output,   is the sensed output,   is the exogenous input, and   is the actuator input, at any  .

The lower linear fractional transformation (LFT) is used to implement a controller   into the system. The lower LFT is denoted as   and is formed by   with  . For full-state feedback we consider a controller of the form  . This is a special case where   and results in a controller of the form  .

The DataEdit

 ,  ,  ,  ,  ,  ,  ,  ,   are known.

The LMI:Full State Feedback Optimal Control LMIEdit

The following are equivalent.

1) There exists a   such that  

2) There exists   and   such that

 .

Then  .

Conclusion:Edit

The above LMI, if feasible, will determine the bound   on the   norm of the system. In addition to this   is also determined allowing the closed loop system to be determined using the controller   found during the optimization.

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_Hinf.m

Related LMIsEdit

Full State Feedback Optimal H2 LMI

External LinksEdit

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